arXiv:1812.02491v1 [math.DS] 6 Dec 2018
HOLOMORPHIC VECTOR FIELDS TANGENT TO FOLIATIONS IN DIMENSION THREE ´ ´ DANUBIA JUNCA & ROGERIO MOL Abstract. This article studies germs of holomorphic vector fields at (C3 , 0) that are tangent to holomorphic foliations of codimension one. Two situations are considered. First, we assume hypotheses on the reduction of singularities of the vector field — for instance, that the final models belong to a family of vector fields whose eigenvalues of the linear part satisfy a condition of non-resonance — in order to conclude that the foliation is of complex hyperbolic type, that is, without saddle-nodes in its reduction of singularities. In the second part, we prove that a vector field that is tangent to three independent foliations is tangent to a whole pencil of foliations — hence, to infinitely many foliations — and, as a consequence, it leaves invariant a germ of analytic surface. This final part is based on a local version of a well-known characterization of pencils of foliations of codimension one in projective spaces.
1. Introduction The main goal of this article is to study, at the origin of C3 , germs of holomorphic vector fields that are tangent to holomorphic foliations of codimension one. If X as a germ of holomorphic vector field at (C3 , 0), inducing a germ of singular holomorphic foliation of dimension one F, and ω is a germ of holomorphic 1−form which satisfies the integrability condition — ω ∧ dω = 0 — inducing a germ of singular holomorphic foliation of codimension one G, we say that X (or F) is tangent to ω (or to G) if the orbits of X are entirely contained in the two-dimensional leaves of ω, wherever both objects are defined. We also say that ω (or G) is invariant by X (or by F). In algebraic terms, this is identified by the vanishing of the contraction of ω by X, that is, iX ω = 0. One interesting point is that, due to the the integrability condition of ω, not every germ of holomorphic vector field at (C3 , 0) is tangent to a homomorphic foliation. Examples of this situation are presented in two recent studies where the configuration of tangency between a vector field and a foliation is considered. The first one, by F. Cano and C. Roche [6], asserts that a germ of holomorphic vector field X at (C3 , 0) tangent to a foliation has a reduction of singularities. This means that, after a finite sequence of blow-ups with invariant centers (points or regular invariant curves in general position with the reduction divisor), the one-dimensional foliation induced by X is transformed into one for which all singularities are elementary, meaning that they are locally defined by vector fields with non-nilpotent linear part. Vector fields in a family proposed by F. Sanz and F. Sancho (see also [6]) do not admit a reduction of singularities as above and, hence, are not tangent 1
2010 Mathematics Subject Classification: 32S65, 37F75 Keywords. Holomorphic foliations, holomorphic vector fields, pencil of foliations 3 First author supported by a CAPES Sandwich Doctorate scholarship. Second author supported by Universal/CNPq. 2
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to any germ of holomorphic foliation. In the second one, by D. Cerveau and A. Lins Neto [8], it is proved that a germ of holomorphic vector field with isolated singularity at 0 ∈ C3 that is tangent to a holomorphic foliation always admits a separatrix, that is, an invariant analytic curve. As a consequence, vector fields in the family of X.G´omex-Mont and I. Luengo [13], which do not posses separatrices, are not tangent to holomorphic foliations. The afore mentioned studies suggest that consequences of geometric nature arise when there is tangency between a vector field and a foliation. This perception is the main motivation for this article and shall be developed in two different and independent approaches. First, in Section 3 we present the concept of strongly diagonalizable germ of vetor field (Def. 3.1), meaning that its linear part has eigenvalues that do not satisfy any non-trivial relation of linear dependency with integer coefficients. We characterize the 1−forms that are tangent to vector fields of this type (Prop. 3.5). In the main result of the section, we consider a germ of integrable holomorphic 1−form ω at (C3 , 0) that leaves invariant a germ of holomorphic vector field X, putting the following hypotheses on the reduction of singularities of X (which exists by Cano-Roche’s result): it is composed only by punctual blow-ups (we say in this case X has an absolutely isolated singularity 0 ∈ C3 ), the divisor associated to this sequence of blow-ups is invariant by the transformed foliation (that is, X is non-dicritical) and that all final models are strongly diagonalizable singularities. Under these assumptions on X, we prove in Theorem 3.7 that ω defines a foliation of codimension one which is complex hyperbolic — this notion is an extension for foliations of codimension one in higher dimensions of the widely studied concept of generalized curve foliations in dimension two (see definitions in Section 2). In Section 4, we investigate the situation where a germ of holomorphic vector field at (C3 , 0) is tangent to three independent foliations, induced by germs of integrable holomorphic 1−forms ω1 , ω2 and ω3 . We prove that, in this case, up to multiplication by germs of functions in O3 , these 1−forms define a pencil of integrable 1−forms, that is, a two-dimensional linear space — which becomes one-dimensional when projectivized — in the space of integrable holomorphic 1−forms. As a consequence, X is tangent to the infinitely many integrable 1−forms in this pencil. In Theorem 4.5, we present a geometric characterization of pencil of integrable 1−forms, stated in the more general context of foliations at (Cn , 0), n ≥ 3, which is a local version of the one given by D. Cerveau in [7]. It asserts that there is a closed meromorphic 1−form θ such that dω = θ ∧ ω for all 1−forms ω in the pencil or the axis foliation— that is, the unique foliation of codimension two that is tangent to all 1−forms in the pencil — has a meromorphic first integral. As a consequence, the axis foliation always admits an invariant hypersurface. This, translated into our original three-dimensional context, gives us that a germ of vector field at (C3 , 0) that is tangent to three independent germs of holomorphic foliations leaves invariant a germ of analytic surface. This paper contains partial results of the Ph.D thesis of the first author. She is thanks N. Corral and the University of Cantabria for hospitality during the development of part of this research.
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2. Preliminaries In local analytic coordinates (x1 , x2 , x3 ) at (C3 , 0), we denote a germ of holomorphic vector field at (C3 , 0) by ∂ ∂ ∂ +B +C , X=A ∂x1 ∂x2 ∂x3 where A, B, C ∈ O3 . We also consider the germ of singular one-dimensional foliation F whose leaves are the orbits of X. Thus, in order to avoid superfluous singularities, we suppose that A, B and C are without common factors. The singular sets (of X or F) are denoted by Sing(X) = Sing(F) = {A = B = C = 0}, being an analytic set of dimension at most one. We denote a germ of of holomorphic 1−form at (C3 , 0) by ω = adx1 + bx2 + cdx3 , where a, b, c ∈ O3 , which are also supposed to be without common factors. If ω is integrable in the sense of Frobenius, that is, ω ∧ dω = 0, it induces a germ of singular holomorphic foliation of codimension one, denoted by G. We have Sing(ω) = Sing(G) = {a = b = c = 0}, also an analytic set of dimension at most one. A separatrix for a local holomorphic foliation is an irreducible germ of invariant analytic variety of the same dimension of the foliation. For germs of holomorphic vector fields and integrable 1−forms at (C3 , 0), separatrices are germs of invariant curves and surfaces, respectively. We say that X (or F) is tangent to ω (or to G) if iX ω = aA + bB + cC = 0, which is equivalent to saying that, outside Sing(X)∪Sing(ω), the orbits of X are contained in the two-dimensional leaves of G. If the germ of vector field X is tangent to the integrable 1−form ω, then Sing(ω) is invariant by X (see, for instance, [16, Th. 1]). Thus, the onedimensional components of Sing(ω) — if they exist — are separatrices of X. One particular example of this configuaration of tangency is provided by vector fields with first integrals. A non-constant germ of meromorphic — or holomorphic — function Φ at (C3 , 0) is a first integral for X if, in a small neighborhood of 0 ∈ C3 , Φ is constant along the orbits of X. This is equivalent to saying that X is tangent to the foliation defined by Φ, which is induced by the holomorphic 1−form obtained by cancelling the components of zeros and poles of the meromorphic 1−form dΦ (we also say that Φ is a first integral for this foliation). Note that in this case, any fiber of Φ accumulating to the origin is an Xinvariant surface. We should mention that, in the article [17], the authors study germs of vector fields at (C3 , 0) that are completely integrable — that is, that have two independent holomorphic first integrals — and prove that this property is not a topological invariant. In dimension two, a germ of holomorphic foliation is induced in local analytic coordinates (x1 , x2 ) at (C2 , 0) by a germ of holomorphic vector field X = A∂/∂x1 +B∂/∂x2 with isolated singularity at the origin (or by the dual 1−form ω = Bdx1 − Adx2 ). Recall that such a foliation has a reduction of singularities, that is, a finite sequence of blow-ups transforms it into one having a finite number of singularities which are simple or reduced [18]. Such a singularity is locally induced by a vector field whose linear part is non-nilpotent, having two eigenvalues λ1 and λ2 such that, if both non-zero, do not satisfy any non-trivial relation of the kind m1 λ1 + m2 λ2 = 0 with m1 , m2 ∈ Z≥0 . These simple singularities are called non-degenerate, whereas the ones having one zero eigenvalue are called saddle-nodes. A foliation at (C2 , 0) is said to be of generalized curve type [1] if it has no saddle-nodes
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in some (and hence in any) reduction of singularities. Several geometric properties of a foliation of generalized curve type can be read in its separatrices. For example, a sequence of blow-ups that desingularizes its set of separatrices is also a reduction of singularities for the foliation itself [1]. A foliation of codimension one at (C3 , 0) also admits a reduction of singularities [4, 3]. This means that after a finite sequence of blow-ups with invariant centers — points and regular curves satisfying a condition of general position with respect to the divisor —, the foliation is transformed into one whose singularities are all simple or reduced, which, in analogy with the two-dimensional case, are essentially of two kinds: simple complex hyperbolic singularities and simple saddle-node singularities (see, for instance, the description in [10]). Recall that the dimensional type of a codimension one foliation, τ ≥ 1, is the smallest number of variables needed to express its defining equation in some system of analytic coordinates. We say that a simple singularities is of complex hyperbolic type [5] if there are analytic coordinates (x1 , x2 , x3 ) at 0 ∈ C3 in which the foliation is defined by a holomorphic 1−form ω whose terms of lowest order are: dx2 dx1 + λ2 , if τ = 2, • x1 x2 λ1 x1 x2 dx1 dx2 dx3 • x1 x2 x3 λ1 + λ2 + λ3 , if τ = 3, x1 x2 x3 where the residues λi ∈ C∗ are non-resonant, that is there are no non-trivial relations of the kind m1 λ1 + m2 λ2 = 0 (for τ = 2) or m1 λ1 + m2 λ2 + m3 λ3 = 0 (for τ = 3), with mi ∈ Z≥0 . Simple complex hyperbolic singularities, when τ = 2, correspond to simple non-degenerate singularities of foliations in dimension two. Now, a foliation G at (C3 , 0) induced by an integrable holomorphic 1−form ω is of complex hyperbolic type if it satisfies one of the two equivalent properties [5]: • There exists a complex hyperbolic reduction of singularities for G, that is, one for which all final models are simple complex hyperbolic. In this case, every reduction of singularities of G will be complex hyperbolic. • for every holomorphic map φ : (C2 , 0) → (C3 , 0) generically transversal to G (that is, such that φ∗ ω has an isolated singularity at 0 ∈ C2 ) the foliation π ∗ F induced by φ∗ ω is of generalized curve type. Complex hyperbolic foliations are the three-dimensional counterparts of generalized curve foliations. It is thus expected that some geometric properties enjoyed by the latter also have a formulation for the former. For instance, in the non-dicritical case (that is, if the reduction divisor is invariant by the transformed foliation), its proved in [12] that a complex hyperbolic foliation becomes reduced once its set of separatrices is desingularized. 3. Strongly diagonalizable vector fields In this section we introduce the notion of strongly diagonalizable germs of vector fields. A vector field in this family has a linear part with eigenvalues satisfying a hypothesis of non-resonance, being, as a consequence, linearizable in formal coordinates. We show that by assuming hypothesis on the reduction of singularities of a germ of vector field at (C3 , 0) that is tangent to a holomorphic foliation of codimension one — among them, that the
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final models belong to this family of strongly diagonalizable vector fields — we are able to conclude that the foliation is of complex hyperbolic type. Recall that a vector α = (α1 , · · · , αn ) ∈ Cn , n ≥ 2, is non-resonant if there are no relations of the form αj = ℓ1 α1 + · · · + ℓn αn = 0, with ℓ1 , . . . , ℓn ∈ Z≥0 satisfying P n j=1 ℓi ≥ 2. A classical result asserts that a germ of complex analytic vector field X at (Cn , 0) whose associated eigenvalues (i.e. those of its linear part DX(0)) are non-resonant is linearizable in formal coordinates. We remark that these linearizing coordinates can be taken to be analytic if these eigenvalues belong to the Poincar´e domain — i.e. the set of vectors α = (α1 , · · · , αn ) ∈ Cn such that the origin 0 ∈ Cn is not in the convex hull of {α1 , · · · , αn }. In the next definition, we work with the following notion: a vector α = (α1 , · · · , αn ) ∈ Cn is said to be strongly non-resonant if there are no non-trivial relations of the form ℓ1 α1 + · · · + ℓn αn = 0, with ℓ1 , . . . , ℓn ∈ Z. Such a non-trivial relation will be called strong resonance. We have: Definition 3.1. A germ of holomorphic vector field at (Cn , 0), n ≥ 2, is said to be strongly diagonalizable if its associated eigenvalues are strongly non-resonant. We denote the family of strongly diagonalizable vector fields by Xsd . Clearly, a vector field in Xsd satisfies the usual condition of non-resonance, being linearizable in formal coordinates. Further, the associated eigenvalues are nonzero and pairwise distinct, implying that its linear part is diagonalizable. We also say that a germ of one-dimensional foliation F at (Cn , 0) is strongly diagonalizable if it is induced by a vector field in Xsd . Evidently, this definition does not depend on the choice of the vector field in Xsd inducing F. In the sequel, we restrain ourselves to ambient dimension n = 3. Thus, if X is a germ of holomorphic vector field at (C3 , 0) in Xsd , we can take local formal coordinates (x1 , x2 , x3 ) such that (1)
X = α1 x1
∂ ∂ ∂ + α2 x2 + α3 x3 , ∂x1 ∂x2 ∂x3
where α1 , α2 , α3 ∈ C∗ are pairwise distinct. Note that if we choose numbers b1 , b2 , b3 ∈ C, not all of them zero, satisfying α1 b1 + α2 b2 + α3 b3 = 0, then X is tangent to the formal meromorphic 1−form dx2 dx3 dx1 + b2 + b3 . ω = x 1 x 2 x 3 b1 x1 x2 x3 We start by proving a simple lemma: Lemma 3.2. A vector field in X ∈ Xsd has exactly three formal smooth separatrices, which correspond to the coordinate axes in its diagonalized form. Proof. We take X in its diagonal form (1). Suppose, without loss of generality, that an X-invariant curve γ is parametrized as γ(t) = (at + f (t), g(t), h(t)) where a ∈ C∗ and ˆ1 are non-units, with f of order at least two. We have to prove that g = h = 0. f, g, h ∈ O Suppose, for instance, g 6= 0. The condition of invariance is expressed as Φ(t)(a + f ′ (t), g′ (t), h′ (t)) = (α1 (at + f (t)), α2 g(t), α3 h(t)),
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ˆ 1 of the form Φ(t) = α1 t + ρ(t), where ν0 (ρ) ≥ 2. The above equation gives for some Φ ∈ O us g ′ (t) α2 = . Φ(t) g(t) Comparing residues in this formula, we find α2 /α1 = m, where m = ν0 (g). This, however, gives a strong resonance for the vector (α1 , α2 , α3 ), in contradiction with our hypothesis. Therefore g = 0 and, in a similar way, h = 0, giving that γ is contained in the x1 -axis. Blow-ups preserve the family of strongly diagonalizable vector fields. More precisely, we have the following: Lemma 3.3. Let X ∈ Xsd . Then the strict transform of the one-dimensional foliation induced by X by a blow-up with smooth invariant center is locally given by vector fields in Xsd . Proof. Take formal diagonalizing coordinates for X. For a punctual blow-up at 0 ∈ C3 , consider coordinates x∗1 = x1 , x∗2 = x2 /x1 and x∗3 = x3 /x1 . In these coordinates, the strict transform of X is e = α1 x∗ ∂ + (α2 − α1 )x∗ ∂ + (α3 − α1 )x∗ ∂ , X 2 3 1 ∂x∗1 ∂x∗2 ∂x∗3
having an isolated singularity at (x∗1 , x∗2 , x∗3 ) = (0, 0, 0). We only have to check that the e are strongly non-resonant. However, a relation of the kind eigenvalues of X 0 = c1 α1 + c2 (α2 − α1 ) + c3 (α3 − α1 ) = (c1 − c2 − c3 )α1 + c2 α2 + c3 α3 ,
for c1 , c2 , c3 ∈ Z, is possible if and only if c1 = c2 = c3 = 0, since the eigenvalues of X are strongly non-resonant. In the case of a monoidal blow-up, its smooth invariant center must be one of the coordinate axes, by Lemma 3.2. For instance, fixing the x3 -axis as the blow-up center and taking blow-up charts x1 = x∗1 , x2 = x∗2 and x∗3 = x3 /x2 , the strict transform of X is ˜ = α1 x∗ ∂ + α2 x∗ ∂ + (α3 − α2 )x∗ ∂ . X 1 2 3 ∂x∗1 ∂x∗2 ∂x∗3 ˜ follows from that of Again, the absence of strong resonances for the eigenvalues of X X. As a consequence, we have: Corollary 3.4. The only formal separatrices of a vector field in Xsd are those corresponding to the coordinate axes in its diagonal form. Proof. Let γ be a formal separatrix for X ∈ Xsd . If γ is smooth, this has been proved in Lemma 3.2. If γ is singular, we desingularize it through a sequence of punctual blowups. By the previous lemma, the strict transform of the foliation induced by X has local models in Xsd . Its smooth invariant curves are either in the desingularization divisor or are contained in the strict transforms of the coordinate axis in diagonalizing coordinates for X. The transform of γ is evidently not in the desingularization divisor. This means that γ is smooth contained in one of the coordinate axes, which is not our case. Our objective now is to prove the following result:
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Proposition 3.5. Let ω be a germ of integrable holomorphic 1−form at (C3 , 0) with codim Sing(ω) ≥ 2. Suppose that ω is invariant by a vector field in X ∈ Xsd . Then, in ˆ3 , we have formal diagonalizing coordinates for X and up to multiplication by a unit in O either dx2 dx1 + b2 (I) ω = x 1 x 2 b1 x1 x2 or dx2 dx3 dx1 + b2 + b3 (II) ω = x 1 x 2 x 3 b1 , x1 x2 x3 where b1 , b2 , b3 ∈ C∗ . Proof. Fix (x1 , x2 , x3 ) formal diagonalizing coordinates for X as in (1) and write (2)
ω = adx1 + bdx2 + cdx3 ,
ˆ3 are without common factors. Since X is tangent to ω, the contraction where a, b, c ∈ O of ω by X gives (3)
0 = iX ω = α1 x1 a + α2 x2 b + α3 x3 c.
The integrability condition in its turn reads (4)
0 = ω ∧ dω = a(cx2 − bx3 ) + b(−cx1 + ax3 ) + c(bx1 − ax2 ).
The differentiation of (3) with respect to each of the variables x1 , x2 and x3 produces the following set of equations: (5) (6) (7)
α1 a + α1 x1 ax1 + α2 x2 bx1 + α3 x3 cx1 = 0; α1 x1 ax2 + α2 b + α2 x2 bx2 + α3 x3 cx2 = 0; α1 x1 ax3 + α2 x2 bx3 + α3 x3 cx3 + α3 c = 0.
We have the following (this was shown to us by M. Fern´ andez-Duque): Assertion 1. In the above conditions, X leaves invariant each ratio of coefficients of ω. Proof of the Assertion. In fact, b2 X (a/b) = bX(a) − aX(b) = b(α1 x1 ax1 + α2 x2 ax2 + α3 x3 ax3 ) − a(α1 x1 bx1 + α2 x2 bx2 + α3 x3 bx3 ) = b(−α1 a − α2 x2 bx1 − α3 x3 cx1 ) + bα2 x2 ax2 + bα3 x3 ax3 − aα1 x1 bx1 −aα2 x2 bx2 − aα3 x3 bx3 (by (6)) = bα3 x3 (ax3 − cx1 ) − abα1 − bα2 x2 bx1 + bα2 x2 ax2 − aα1 x1 bx1 −aα2 x2 bx2 − aα3 x3 bx3 = bα3 x3 (ax3 − cx1 ) + aα3 x3 (cx2 − bx3 ) + ab(α2 − α1 ) +bα2 x2 (ax2 − bx1 ) + aα1 x1 (ax2 − bx1 ) (by (7)) = α3 x3 (c(ax2 − bx1 )) + ab(α2 − α1 ) + (ax2 − bx1 )(aα1 x1 + bα2 x2 ) (by (4)) = ab(α2 − α1 ) + (ax2 − bx1 )(α1 ax1 + α2 bx2 + α3 cx3 ) (by (3)) = ab(α2 − α1 ).
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That is, X (a/b) = (α2 − α1 )a/b. In a similar way, we find X (a/c) = (α3 − α1 )a/c and X (b/c) = (α3 − α2 )b/c, proving the assertion. We have just found that X(a/b) = µ1 a/b,
X(a/c) = µ2 a/c
and
X(b/c) = µ3 b/c,
where µ1 = α2 − α1 , µ2 = α3 − α1 and µ3 = α3 − α2 are in C∗ . These equations can be rewritten as (8)
bX(a) − aX(b) = µ1 ab,
cX(a) − aX(c) = µ2 ac and cX(b) − bX(c) = µ3 bc.
The first of these equations is equivalent to bX(a) = a(X(b) + µ1 b), where we can see that the factors of a that do not divide b do divide X(a). Similarly, from second equation we have cX(a) = a(X(c) + µ2 c), allowing us to conclude that the factors of a that do not divide c do divide X(a). Since a, b and c do not have common factors we conclude that a divides X(a). Analogously, b divides X(b) and c divides X(c). Therefore, we can find ˆ3 such that functions R1 , R2 , R3 ∈ O X(a) = R1 a,
X(b) = R2 b
and
X(c) = R3 c.
From equations (8) we have R1 − R2 = µ1 ,
R1 − R3 = µ2
and
R2 − R3 = µ3 .
ˆ3 is a non-unity. From For i = 1, 2, 3, we write Ri = (λi + fi ), where λi ∈ C and fi ∈ O the above equations, we have λ1 − λ2 = µ1 = α2 − α1 ,
λ1 − λ3 = µ2 = α3 − α1 ,
λ2 − λ3 = µ3 = α3 − α2
and f1 = f2 = f3 = f . Suppose that a 6= 0 and denote by aν be its initial part, that is, the homogeneous part of order ν = ν0 (a) of its Taylor series. Taking initial parts in both sides of X(a) = (λ1 + f )a and considering the fact that the derivation by X preserves the degree — actually, the multidegree — of each monomial, we have X(aν ) = λ1 aν . Further, if κ = b1 xi1 xj2 xk3 is a non-zero monomial in aν , where b1 ∈ C∗ , then X(κ) = λ1 κ, which gives λ1 = iα1 + jα2 + kα3 6= 0. Note that the fact that (α1 , α2 , α3 ) is strongly non-resonant implies that aν = κ. Assertion 2. aν divides a. P Proof of the Assertion. Write the power series a = ℓ≥ν aℓ , where aℓ assembles the homogeneous terms of degree ℓ. We will show by induction that aν = b1 xi1 xj2 xk3 divides each aℓ . There is nothing to prove for ℓ = ν. Let m > ν and suppose that aℓ is divisible by aν for all ℓ = ν, . . . , m − 1. Let ̺ be a monomial of am . We have X(̺) = λ̺ for some λ ∈ C. Considering the calculation in the above paragraph, since λ1 has already been determined by the multidegree of aν , we must have λ 6= λ1 . On the other hand, separating all monomials of the same multidegree of ̺ in the expression X(a) = λ1 a + f a, we have (9)
X(̺) = λ1 ̺ + ̺˜,
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where ̺˜ assembles all monomials coming from f a. Notice that ̺˜ can be seen a combination of monomials of a of order smaller than m having monomials of f as coefficients. Hence, by the induction hypothesis, ̺˜ is divisible by aν . Rewriting (9) as λ̺ = λ1 ̺ + ̺˜, we find (λ − λ1 )̺ = ̺˜, from where we deduce that ̺ is also divisible by aν . We then conclude that aν divides am , proving the general step of the induction. Suppose that the coefficients a, b, and c of ω are non-zero. By Assertion 2, we can write a = xi11 xj21 xk31 (b1 + g1 ),
k3 b = xi12 xj22 xk32 (b2 + g2 ) and c = xi13 xj3 2 x3 (b3 + g3 ),
ˆ3 are non-units. Since X is tangent to ω, we have where b1 , b2 , b3 ∈ C∗ and g1 , g2 , g3 ∈ O α1 b1 xi11 +1 xj21 xk31 + α2 b2 xi12 xj22 +1 xk32 + α3 b3 xi13 xj23 xk33 +1 = 0, which implies that i1 + 1 = i2 = i3 ,
j1 = j2 + 1 = j3
and
k1 = k2 = k3 + 1.
Since a, b, c do not have common factors, we find straight that i1 = j2 = k3 = 0, giving i2 = i3 = 1,
j1 = j3 = 1
and
k1 = k2 = 1.
Now we can write
(10)
ω = x2 x3 (b1 + g1 )dx1 + x1 x3 (b2 + g2 )dx2 + x1 x2 (b3 + g3 )dx3 dx2 dx3 dx1 + (b2 + g2 ) + (b3 + g3 ) . = x1 x2 x3 (b1 + g1 ) x1 x2 x3
Dividing equation (10) by 1 + g1 /b1 , we can rewrite, abusing, notation, (11)
ω = b1 x2 x3 dx1 + x1 x3 (b2 + g2 )dx2 + x1 x2 (b3 + g3 )dx3 .
Let us apply the relation X(b/a) = −µ1 b/a of (8) to this writing of ω. Write X x1 b = (b2 + g2 ) = αijk xi1 xj2 xk3 a b1 x 2 i,k≥0, j≥−1
as a sum of meromorphic monomials. Since the derivation by X preserves monomials, we have that α1 i + α2 j + α3 k = −µ1 whenever αijk 6= 0. Again, the fact that (α1 , α2 , α3 ) is free from strong resonances implies that b/a is a monomial. Thus g2 = 0 and b b2 x 1 = , a b1 x 2 implying b = b2 x1 x3 . In an analogous way, we can also prove that c = b3 x1 x2 . This leads to the form (II) in the statement of the proposition. The case where one of the coefficients of ω is zero, for example, c = 0, is treated following the same steps above, giving form (I) in the statement. Proposition 3.6. Let ω be a germ of integrable holomorphic 1−form at (C3 , 0) invariant by a vector field in X ∈ Xsd . Then ω is complex hyperbolic. Proof. We apply Proposition 3.5. If the dimensional type is two, it is straight to see that ω is simple complex hyperbolic. If the dimensional type is three, then, except for a possible resonance of its residues, ω has the form of a simple complex hyperbolic singularity. However, these resonances can be eliminated by punctual or monoidal blow-ups [3, 11],
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obtaining simple singularities of complex hyperbolic type. We then conclude that ω is complex hyperbolic. Next result exemplifies how vector fields and codimension one foliations satisfying a relation of tangency can be geometrically entwined. Before stating it, we set a definition: a germ holomorphic vector field — or its associated one-dimensional holomorphic foliation — at (C3 , 0) has an absolutely isolated singularity at 0 ∈ C3 if it admits a reduction of singularities having only punctual blow-ups. We call the corresponding composition of blow-up maps an absolutely isolated reduction of singularities. Theorem 3.7. Let F be a germ of one-dimensional holomorphic foliation at (C3 , 0) admitting a non-dicritical absolutely isolated reduction of singularities whose associated final models are all strongly diagonalizable. If G is a germ of foliation of codimension one invariant by F, then G is of complex hyperbolic type. Proof. The proof goes by induction on n, the minimal length of all absolutely isolated reductions of singularities for F as in the theorem’s assertion. If n = 0 the result follows from Proposition 3.6. Suppose then that n > 0 and that the result is true for onedimensional foliations having non-dicritical absolutely isolated reductions of singularities with strongly diangonal final models of length less than n. Denote by π : (M, E) → (C3 , 0) the first punctual blow-up of the corresponding reduction of singularities of F. If G1 = π ∗ G were non-singular over the divisor E = π −1 (0) ≃ P2 , then Sing(G) would be an isolated singularity at 0 ∈ C3 . As a consequence of Malgrange’s Theorem [15], in this case G would have a holomorphic first integral, being of complex hyperbolic type. We can then suppose that Sing(G1 ) ∩ E 6= ∅ and pick p ∈ Sing(G1 ) ∩ E. If p ∈ Sing(F1 ), then, by the induction hypothesis, we must have that G1 is of complex hyperbolic type at p. Suppose then that p is regular for F1 . In this case, since F1 is tangent to G1 , the foliation G1 has dimensional type two at p and the leaf of F1 at p is a curve contained in the one-dimensional a analytic set Sing(G1 ). Since E is invariant by F1 , the component of Sing(G1 ) containing this leaf is contained in E and, hence, it is an algebraic curve in E ≃ P2 , that we denote by γ. Now, the sum of Camacho-Sad indices of F1|E along γ is the self-intersection number γ · γ > 0 (see [2, 19]). This assures the existence of a singularity q ∈ Sing(F1|E ), which is obviously a singularity of F1 . By the induction hypothesis, q is of complex hyperbolic type for G1 . In view of this, the transversal model of G1 along (the generic point of) γ is of complex hyperbolic type, leading to the conclusion that G1 is of complex hyperbolic type at p. We have found that each singularity of G1 over E is of complex hyperbolic type, admitting a complex hyperbolic reduction of singularities. This means that G itself has a complex hyperbolic reduction of singularities, being a foliation of complex hyperbolic type. 4. Integrable pencils of 1−forms The goal of this section is characterize the situation in which a germ of holomorphic vector field at (C3 , 0) is tangent to three independent holomorphic foliations. We show that, when this happens, the vector field is tangent to infinitely many foliations and it leaves invariant a germ of analytic surface. To this end, we work with the notion of pencil of integrable 1-forms or pencil of foliations. We will formulate our results in the broader context of holomorphic foliations of codimension one at (Cn , 0), n ≥ 3, that leave invariant foliations of codimension two.
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We start with a definition. Let ω1 and ω2 be independent germs of holomorphic 1−forms at (Cn , 0), that is, such that ω1 ∧ω2 6= 0. The pencil of 1−forms with generators ω1 and ω2 is the linear subspace P = P(ω1 , ω2 ) of the complex vector space of germs of holomorphic 1−forms (Cn , 0) formed by all 1−forms ω(a,b) = aω1 + bω2 , where a, b ∈ C. We have the following lemma, which is a local version of [16, Lem. 2]: Lemma 4.1. If the independent germs of holomorphic 1−forms ω1 and ω2 do not have common components of codimension one in their singular sets, then the 1−forms aω1 + bω2 ∈ P = P(ω1 , ω2 ) have singular sets of codimension at least two, except possibly for a finite number of values (a : b) ∈ P1 . Proof. Suppose, by contradiction, that the result is false. Then, for infinitely many values of t ∈ C, the 1−form ω1 + tω2 ∈ P has some component of codimension one in its singular set, defined by and irreducible P P gt ∈ On . Let us consider these values of t. Writing ω1 = ni=1 Ai dxi and ω2 = ni=1 Bi dxi , where Ai , Bi ∈ On , we have that, for each pair i, j, with 1 ≤ i < j ≤ n, both Ai + tBi = 0 and Aj + tBj = 0 are zero over gt = 0, implying that Ai Bj − Aj Bi = 0 over this same set. However, the fact that Sing(ω1 ) and Sing(ω2 ) do not have a common component of codimension one imply that independent functions gt are associated to different values of t. Hence Ai Bj − Aj Bi ≡ 0 and, consequently, Ai /Bi ≡ Aj /Bj for each pair i, j. By setting Φ = Ai /Bi — which is independent of the chosen i — we have a germ of meromorphic function at (Cn , 0) such that ω1 = Φω2 . This contradicts the fact that ω1 and ω2 are independent 1−forms. Consider a pair of germs of holomorphic 1−forms ω1 and ω2 as in the lemma. We say that P = P(ω1 , ω2 ) is a pencil of integrable 1−forms or an integrable pencil if all its elements are integrable 1−forms, that is, ω ∧ dω = 0 for all ω ∈ P. This is equivalent to the following fact, which will be referred to as pencil condition: (12)
ω1 ∧ dω2 + ω2 ∧ dω1 = 0.
Observe that, after possibly cancelling codimension one components in the singular set, we associate to each ω(a,b) ∈ P a germ of singular holomorphic foliation F t , where t = (a : b) ∈ P1 . For this reason, we also treat this object as pencil of holomorphic foliations. The 2−form ω1 ∧ ω2 is also integrable, defining, after cancelling singular components of codimension one, a singular holomorphic foliation of codimension two which is tangent to all foliations (associated to 1−forms) in P = P(ω1 , ω2 ). This codimension two foliation is called axis of P. Example 4.2. (Logarithmic 1−forms) Take independent irreducible germs of functions f1 , . . . , fk ∈ On , where k ≥ 2. Consider also (λ1 , . . . , λk ) and (µ1 , . . . , µk ) two C-linearly independent k-uples of numbers in C∗ . Then, the holomorphic 1−forms dfk dfk df1 df1 · · · λk · · · µk and ω 2 = f 1 · · · f k µ1 ω 1 = f 1 · · · f k λ1 f1 fk f1 fk are generators of a pencil of integrable 1−forms. The axis foliation is defined by the 2−form X 1 ω1 ∧ ω2 = (λi µj − λj µi )hij dfi ∧ dfj , f1 · · · fk 1≤i 0. Condition (13) then says that each hypersurface fi = 0 in invariant by every ω ∈ P and, hence, also by the axis of P. Suppose, on the other hand, that θ holomorphic. Since dθ = 0, there exists h ∈ On such that θ = dh and hence dω = dh ∧ ω for every ω ∈ P. Setting h′ = exp(h) ∈ O∗n , we have dh′ /h′ = dh and thus dω = (dh′ /h′ ) ∧ ω. Then d (ω/h′ ) = 0, implying that there exists f ∈ On such that ω/h = df . Hence, each foliation in P has a holomorphic first integral, and this implies that the axis of P is completely integrable, that is, it has two independent holomorphic first integrals. In particular, the axis foliation leaves invariant germs of analytic hypersurfaces. Case 2: k(P) 6= 0. Let ω1 and ω2 be generators of the pencil. We have the following: Assertion. There exists a germ of meromorphic function α at (Cn , 0) such that (15)
dθ = αω1 ∧ ω2 .
Proof of the Assertion. First note that, taking differentials in both sides of dωi = θ ∧ ωi , we have dθ ∧ ωi = 0 for i = 1, 2. Now, let q be a point near 0 ∈ Cn such that (ω1 ∧ ω2 )(q) 6= 0. This means that ω1 and ω2 are non-singular and linearly independent at q, so that we can find analytic coordinates (x1 , x2 , · · · , xn ) at q such that ω1 = A1 dx1 and ω2 = A2 dx2 , where P A1 , A2 are invertible germs of holomorphic functions at q. Write dθ = B1,2 dx1 ∧ dx2 + (i,j)6=(1,2) Bi,j dxi ∧ dxj , where each Bi,j is a germ of the meromorphic function at q. Since dθ ∧ ω1 = 0 and dθ ∧ ω2 = 0, we must have Bi,j = 0 whenever (i, j) 6= (1, 2). Then, at q, B1,2 (A1 dx1 ) ∧ (A2 dx2 ) = αω1 ∧ ω2 , dθ = B1,2 dx1 ∧ dx2 = A1 A2 where α is a germ of the meromorphic function at q. In this way, we produce a meromorphic function α, defined outside the set of tangencies Tang(ω1 , ω2 ), which, by comparing coefficients of dθ and ω1 ∧ω2 as we did in the proof of Proposition 4.3, can be extended to a meromorphic function defined in a neighborhood of 0 ∈ Cn . This proves the assertion. Now, we split our analysis in two subcases: Subcase 2.1: α is constant. We claim that the axis foliation of P is tangent to the levels of a meromorphic (possibly holomorphic) function. Indeed, α 6= 0, since k(P) 6= 0, so that, from the exterior derivative of (15), we get that dω1 ∧ ω2 − ω1 ∧ dω2 = 0. This, together with the pencil condition (12), gives ω2 ∧ dω1 = ω1 ∧ dω2 = 0. Hence, using (13), we find that θ ∧ ω1 ∧ ω2 = 0. Applying Proposition 4.3 to the 1−forms ω1 , ω2 and ω3 = θ and to the 2−form η = ω1 ∧ ω2 , we find germs of meromorphic functions µ1 and µ2 at (Cn , 0) satisfying θ = µ1 ω 1 + µ2 ω 2 . Inserting this in (13), we get dω1 = −µ2 ω1 ∧ ω2
and
dω2 = µ1 ω1 ∧ ω2 .
If µ1 = 0 then dω2 = 0, and then ω2 = d(g) for some g ∈ On , which turns out to be a holomorphic first integral for the axis of P. On the other hand, when µ1 6= 0, the above
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equations give dω1 = −(µ2 /µ1 )dω2 , which, by differentiation, yields d(µ2 /µ1 ) ∧ ω1 ∧ ω2 = 0. When µ2 /µ1 is non-constant, we have at once that µ2 /µ1 is a meromorphic first integral for the axis of P. When µ2 /µ1 = c for some c ∈ C, we have dω1 = −cdω2 . That is to say, ω1 + cω2 is a 1−form in P which is closed, and hence exact, yielding once again a holomorphic first integral for the axis foliation. Subcase 2.2: α is non-constant. Taking the exterior derivative of (15) and using (13), we obtain dα (16) + θ ∧ ω1 ∧ ω2 = 0. 2α Now, applying Proposition 4.3 to ω1 , ω2 , ω3 = dα/2α + θ and η = ω1 ∧ ω2 , we can find k1 and k2 , germs of meromorphic functions at (Cn , 0), such that dα + θ = k1 ω1 + k2 ω2 . (17) 2α Observe that, since k(P) 6= 0, we have that either k1 or k2 is non-zero. Taking the exterior derivative and applying (13), we obtain dθ = (k1 θ + dk1 ) ∧ ω1 + (k2 θ + dk2 ) ∧ ω2 . This, wedged by ω1 and ω2 , gives, respectively, dk2 (18) θ+ ∧ ω1 ∧ ω2 = 0 and k2 Subtracting (16), we obtain, respectively, 1 dα dk2 + ∧ ω1 ∧ ω2 = 0 (19) − 2 α k2
and
dk1 θ+ k1
∧ ω1 ∧ ω2 = 0.
1 dα dk1 + − 2 α k1
∧ ω1 ∧ ω2 = 0.
This allows us to conclude that he meromorphic functions k12 /α and k22 /α are constant on the leaves of the axis foliation of P. If one of these two functions is non-constant, we have a meromorphic first integral for the axis foliation and the proof of the theorem is accomplished. We claim that this actually happens. Indeed, suppose, by contradiction, that both k12 /α and k22 /α are constant. This would imply that k1 /k2 (if k2 6= 0) is also constant. Writing k1 /k2 = c1 ∈ C, we get, from (17), dα + θ = k2 (c1 ω1 + ω2 ) (20) 2α and, since k22 /α is constant, we find that dk2 (21) θ=− + k2 (c1 ω1 + ω2 ). k2 Then, applying (13) to c1 ω1 + ω2 ∈ P , we find dk2 (22) d(c1 ω1 + ω2 ) = − ∧ (c1 ω1 + ω2 ). k2 This gives that k2 (c1 ω1 + ω2 ) is closed. By (21), we would have k(P) = dθ = 0, reaching a contradiction. Let us put the previous discussion our initial three-dimensional context:
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Theorem 4.6. Let X be a germ of holomorphic vector field at (C3 , 0) tangent to three independent foliations of codimension one. Then there exists an integrable pencil P such that X is tangent to all foliations in P. Furthermore, at least one of the following two conditions holds: i) there exists a closed meromorphic 1−form θ such that dω = θ ∧ ω for every ω ∈ P; ii) X has a non-constant meromorphic (possibly holomorphic) first integral. In particular, there exists a germ of analytic surface at 0 ∈ C3 which is invariant by X. We close this article with an example. Example 4.7. (Jouanolou’s Example) Consider, for m ≥ 2, the vector field X = m m xm 3 ∂/∂x1 + x1 ∂/∂x2 + x2 ∂/∂x3 . Then ω = iR iX Ω, where Ω = dx1 ∧ dx2 ∧ dx3 is the volume form and R = x1 ∂/∂x1 + x2 ∂/∂x2 + x3 ∂/∂x3 is the radial vector field, is an integrable 1−form, with homogeneous coefficients, that is invariant by X. Since iR ω = 0, ω defines a foliation on the complex projective plane P2C which leaves invariant no algebraic curve [14]. This is equivalent to saying that, at 0 ∈ C3 , the vector vector field X leaves invariant no homogeneous surface — that is, a surface defined by the vanishing of a homogeneous polynomial in the coordinates (x1 , x2 , x3 ). Since X itself is a homogeneous vector field, this implies that X is not tangent to any germ of analytic surface at 0 ∈ C3 . By Theorem 4.6, X is cannot be tangent to three independent foliations. References [1] C. Camacho, A. Lins Neto, and P. Sad. Topological invariants and equidesingularization for holomorphic vector fields. J. Differential Geom., 20(1):143–174, 1984. [2] C. Camacho and P. Sad. Invariant varieties through singularities of holomorphic vector fields. Ann. of Math. (2), 115(3):579–595, 1982. [3] F. Cano. Reduction of the singularities of codimension one singular foliations in dimension three. Ann. of Math. (2), 160(3):907–1011, 2004. [4] F. Cano and D. Cerveau. Desingularization of nondicritical holomorphic foliations and existence of separatrices. Acta Math., 169(1-2):1–103, 1992. [5] F. Cano, M. Ravara-Vago, and M. Soares. Local Brunella’s alternative I. RICH foliations. Int. Math. Res. Not. IMRN, (9):2525–2575, 2015. [6] F. Cano and C. Roche. Vector fields tangent to foliations and blow-ups. J. Singul., 9:43–49, 2014. [7] D. Cerveau. Pinceaux lin´eaires de feuilletages sur CP(3) et conjecture de Brunella. Publ. Mat., 46(2):441–451, 2002. [8] D. Cerveau and A. Lins Neto. Codimension two holomorphic foliations. To appear: J. Differential Geom., 2018. [9] D. Cerveau and J.-F. Mattei. Formes int´egrables holomorphes singuli`eres, volume 97 of Ast´erisque. Soci´et´e Math´ematique de France, Paris, 1982. [10] G. Cuzzuol and R. Mol. Second type foliations of codimension one. Bol. Soc. Mat. Mex., doi.org/10.1007/s40590-018-0207-9, 2018. [11] M. Fern´ andez Duque. Elimination of resonances in codimension one foliations. Publ. Mat., 59(1):75–97, 2015. [12] P. Fern´ andez-S´ anchez and J. Mozo-Fern´ andez. On generalized surfaces in (C3 , 0). Ast´erisque, (323):261–268, 2009. [13] Xavier G´ omez-Mont and Ignacio Luengo. Germs of holomorphic vector fields in C3 without a separatrix. Invent. Math., 109(2):211–219, 1992. ´ [14] J. P. Jouanolou. Equations de Pfaff alg´ebriques, volume 708 of Lecture Notes in Mathematics. Springer, Berlin, 1979. ´ [15] B. Malgrange. Frobenius avec singularit´es. I. Codimension un. Inst. Hautes Etudes Sci. Publ. Math., (46):163–173, 1976.
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[16] R. Mol. Flags of holomorphic foliations. Anais da Academia Brasileira de Ciˆencias, 83(3):775–786, 2011. [17] S. Pinheiro and H. Reis. Topological aspects of completely integrable foliations. J. Lond. Math. Soc. (2), 89(2):415–433, 2014. [18] A. Seidenberg. Reduction of singularities of the differential equation A dy = B dx. Amer. J. Math., 90:248–269, 1968. [19] T. Suwa. Indices of holomorphic vector fields relative to invariant curves on surfaces. Proc. Amer. Math. Soc., 123(10):2989–2997, 1995.
Dan´ ubia Junca Departamento de Matem´atica Universidade Federal de Itajub´ a Rua Irm˜a Ivone Drumond, 200 35903-087 Itabira - MG BRAZIL
[email protected]
Rog´erio Mol Departamento de Matem´atica Universidade Federal de Minas Gerais Av. Antˆonio Carlos, 6627 C.P. 702 30123-970 - Belo Horizonte - MG BRAZIL
[email protected] 
